Denver Office August 1988 - Bureau of Reclamation · REC-ERG88-3 ‘ l I OVERTOPPING FLOW ON LOW EMBANKMENT DAMS - SUMMARY REPORT OF MODEL TESTS by R. A. Dodge August 1988 Hydraulics

Denver Office

August 1988

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; U. S. Department of the Interior t

Bureau of Reclamation .

79090 (4.91) Burem ot RsolMUt- TECl4NlP.A I RC iPORT STANDARD TITLE PAGE

3. RECIPIENT’S CATALOG NO.

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I >. IDENTIFIERS-- Blue Ridge Parkway Dam/

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Overtopping Flow on Low Embankment Dams - Summary Report of Model Tests

7. AUTHOR(S)

R. A. Dodge

9. PERFORMING ORGANlZATlON NAME AND AOORESS

Bureau of Reclamation Denver Off ice Denver, Colorado

Tz. SPONSORING AGENCY NAME ANO ADDRESS

Same

IS. SUPPLEMENTARY NOTES

5. REPORT DATE

August 1988 6. PERFORMING ORGANIZATION CODE

D-1531

9. PERFORMING ORGANIZATION REPORT NO.

REC-ERC-883

10. WORK UNIT NO.

II. CONTRACT OR GRANT NO.

13. TYPE OF REPORT AND PERIOD COVERED

14. SPONSORING AGENCY CODE

DIBR

Microfiche and/or hard copy available at the Denver Office, Denver, Colorado

16. ABSTRACT

Ed:RNW

A 1: 15 scale model was used to study the relative effectiveness of different embankment protective treatments. Tests were done representing unit discharges of 40 and 86 ftJ/s. Chute and pool mode flow occurred for all treatments tested. Chute and pool flow had less scouring power compared with the earlier less rough plane flow. A smooth hard crest cap going part way down the slope caused a deep scour hole just downstream of the cap. The same flow, after roughening the sloping part of the cap with epoxied pea gravel simulating fixed cobble roughness, scoured one-hatf the volume compared with plain cap test. Riprap, with a simulated maximum size of 24 inches, immediately fluidiied and washed out completely. There was no indication that any of the mesh-contained treatments would fail. Flow on an embankment slope of 4: 1, protected by gabbions scoured 2.5 times the volume for a gabbion protected 1:6 slope. A change of compaction from 95 to 102 percent standard Proctor density decreased scour one-half. Doubling the unit discharge increased scour by a factor of 1.4 times. Both design and modeling are hampered by lack of sufficiently verified equations for steep flow and for chute and pool flow. Therefore these test results are relative and future research is suggested.

17. KEY WORDS AND DOCUMENT ANALYSIS

3. DESCR JPTORS-- hydraulic models/ ‘overtopping flow/ embankments/ *erosion/ low dams/ ‘bank protection/ plain soil/ extended crest caps/ roughened cap/ gabbions/ rock mattress/ riprap

c. COSATI Field/Group 08M COWRR: 0813 SRIM:

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REC-ERG88-3

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l

I OVERTOPPING FLOW ON LOW EMBANKMENT DAMS -

SUMMARY REPORT OF MODEL TESTS

by R. A. Dodge

August 1988

Hydraulics Branch Research and laboratory Services Division Denver Off ice Denver, Colorado

UNITED STATES DEPARTMENT OF THE INTERIOR * BUREAU OF RECLAMATION

The success of the model studies is mainly attributed to the exceptional performance and dedication of rotation engineers, Tom E. Sadusky and Betty J. Chavira who helped place soil, supervised test runs and analyzed data. The continued meeting of tight time schedules by Kathy A. lnman of our laboratory in preparing and placing the soil is greatly appreciated. Jack G. Byers, Jack Rosenfield, and Kurt Von Fay of the Geotechnical Branch found a local source of soil and provided continued assistance in monitoring the compaction of the model embankments.

George Powledge was Chief, Dam Safety Support Branch when the study was requested. He provided direct support and input for all the study. After Harley Warren (retired) became Chief of the Dam Safety Support Branch, he continued the same support and input. The background part of this report was essentially written by George Powledge (coauthor [ 11) now Chief of the Bureau’s Program Management office.

The laboratory studies were supervised by T. J. Rhone, Head - Hydraulic Structures Section of the Hydraulics Branch. P. H. Burgi was Hydraulics Branch Chief during these studies.

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As the Nation’s principal conservation agency, the Department of the Interior has responsibility for most of our nationally omed public lands and natural resources. This indudes fostering the wisest use of our land and water resources, protecting our fish and wildlife, preserv- ing the environmental and cultural values of our national parks and historical places, and providing for the enjoyment of life through out- door recreation. The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The Depamnent also has a major responsibility for American Indian reservation communities and for people who live in Island Territories under U.S. Administration.

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CONTENTS Page

. Introduction ................................................................................................................................. 1 Purpose of the model study ...................................................................................................... 1 Background ............................................................................................................................. 1

Conclusions ................................................................................................................................ 1 Conclusions based on model tests ............................................................................................ Conclusions related to modeling and design needs ..................................................................... :

Technical considerations and analyses ........................................................................................... 2 General scour considerations .................................................................................................... Turbulence .............................................................................................................................. 5 Uniform flow equations for tractive shear caused by flow ........................................................... Limits of uniform flow equations ............................................................................................... : Large scale roughness .............................................................................................................. 5 Secondary flow ....................................................................................................................... 6 Crest hydraulics and crest length .............................................................................................. 6 Boundary roughness changes ................................................................................................... 7 Early threshold tractive shear and velocity concepts.. ................................................................. 7 Relative scour resistance in terms of soil classification and properties ......................................... 8 Entrainment functions ............................................................................................................... 9 Gravity effects on erosion resistance ......................................................................................... 11 Sediment transport functions.. .................................................................................................. 11 General model similitude ........................................................................................................... 17 Hydraulic friction and flow scaling ............................................................................................. 17 Sediment transport rate scaling ................................................................................................ 18

Laboratory model studies ............................................................................................................. Model study program ............................................................................................................... :: Model scaling used for laboratory test runs ............................................................................... 19 Laboratory test facility ............................................................................................................. 19 Soil tested .............................................................................................................................. 19 Model operation ...................................................................................................................... 20 Description of slope treatments tested ...................................................................................... Results .................................................................................................................................... z:

Other possible treatments not tested ............................................................................................ 25

Bibliography ................................................................................................................................ 27

TABLES Table

1 Soil classification with relative erosion resistance ................................................................ 9 2 Comparison of Etchevery’s maximum allowable velocities and tractive forces.. ...................... 11 3 Comparison of Fortier and Scobey’s limiting velocities and tractive force values.. .................. 12 4 U.S.S.R. Limiting velocities and tractive forces in cohesive material ...................................... 12 5 Results of overtopping flow model - summary .................................................................. 23

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Figure

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10 11 12 13 14

CONTENTS - Continued Page

Definition for gradually accelerating flow ............................................................................. 4 Definition for momentum analysis ....................................................................................... 5 Secondary flow cells in flow section ................................................................................... 6 Erosion characteristics for fine-grained cohesive soils with respect to plasticity.. ................... 10 Shields’ diagram for threshold of bed material movement .................................................... 13 Gravity correction for uphill and downhill flow ..................................................................... 14 Angle for repose for rock material ...................................................................................... 15 Gessler’s probability modification of Shields’ function .......................................................... 15 Graf and Pazis modification of Shields’ entrainment function ................................................ Taylor’s dimensionless sediment discharge plotted with Shields’ function .............................. :i Laboratory test facility ...................................................................................................... Gradation test of soil used for test runs 1 through 9 ........................................................... 2”1 Compaction-penetration resistance cunres ......................................................................... Typical scour of test embankments ....................................................................................

LETTER SYMBOLS AND QUANTITIES

R ti HC 4 k

9.. 9m. Q

:., Rh

Area of flow section Flow area of the critical depth flow section or control section Wetted area that forces flow between roughness elements Incremental flow area at control station Dimensionless cohesion, function of some combination of soil parameters Percent clay in a soil Compressive strength Subscript denoting soil compaction Subscript denoting characteristic or boundary value Chezy coefficient Sediment diameter or differential operator d is sediment diameter where i equals (or less than) 50, 90, or 100 percentage of the sediment

weight (mass) finer than that diameter Mean depth of flow reach Depth of flow section Flow depth at control section Approach flow depth Dimensionless flow depth Base of natural logrithm, 2.71828 . . . Soil voids ratio Subscript denoting erosion Darcy Weisbach friction factor Steep flow and large element roughness friction factor Friction factor at threshold of sediment motion Froude number, 4&W/v or VJg1/2x1/2 Gravitational constant (acceleration) Friction head loss head Head at critcal or control section Approach head Dimensional erosion constant, same units as sediment discharge Skewness of sediment size distribution Rugosity, boundary surface roughness A mass erodiibilii constant A volume erosion constant An erosion constant Subscript denoting level Crest length Model length scale ratio Subscript denoting mass Manning’s coefficient An exponent Dimensionless parameter Subscript denoting boundary value Flow section wetted perimeter Soil density compaction parameter Discharge per unit width Unit discharge ratio Unit mass sediment discharge Sediment discharge in volume per unit width per second. (In Taylor’s dimensionless discharge

parameter or defined by units used in other empirical equation for cohesive transfer) Dimensionless unit volume discharge Unit volume sediment discharge Water discharge Reynolds number 4 Vr/v Shear velocity grain diameter Reynolds number, U&J Hydraulic radius, A/P

Y2 Z

a a, a1 B

h

? A

Hydraulic radius at threshold of sediment motion Reynolds number distortion ratio Subscript denoting sediment Slope, horizontal to vertical Vane shear strength Slope of energy gradient at threshold of sediment motion Top width of flow at control section Time scale ratio Shear velocity Shear velocity at threshold of sediment movement Velocity of incremental area of control section Average flow velocity Characteristic average velocity of flow Average velocity on a level bed Velocity scale ratio Average velocity on a dam or canal bank Velocity at threshold of sediment motion Dimensionless velocity Velocity at distance of y, above bed Velocity at distance y2 above bed Velocity fluctuation Width of rough element flow section Subscript denoting water Distance along flow slope Characteristic distance along flow slope Dimensionless distance along flow slope A distance above bed below y2 A distance above bed above y, Distance of flow bed above elevation datum

Velocity distribution coefficient or kinetic energy correction factor Control section velocity distribution coefficient Measuring head section velocii distribution coefficient Angle of flow relative to a horizontal line in the plane of canal side slope Specific weight of water, density times gravitstionsl constant Small increment Turbulent intensity Angle of repose for bank material Roughness concentration parameter, a function of spacing, distribution roughness elements, and

element size relative to depth of flow Subscript denoting mean Kinematic viscosity Density of water Density of sediment Trsctive shear on flow bed Standard deviation of grain size distribution Tractive shear on level bed Tractive shear on sloping bank Tractive shear at threshold of sediment motion Dimensionless tractive shear Slope angle canal side bank or dam embankment Function operator

Vi

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INTRODUCTION Purpose of the Model Study

The investigations were to detemrine, with a sectional model, the modes of flow and the erosion on dOwnStre8m Slop8 protection for Small embankment d8mS during overtopping flow. Model experience will greatly aid t0 determine model SCal8S 811d capability, 8nd to decide whether further physical model studies are feasible. The results will be used to evaluate future research needs. These model results can be used to help form concepts for treating existing small embankment dams so they can possibly overtop resulting from the larger design flows that have been determined by contemporary flood routing methods. The studies were supported by the Bureau of Rec- lamation Dam Safety Program concerning overtopping flow on low embankment d8mS.

Background

lt is assumed that when 8n embankment dam is over- topped, erosion on the downstream slop8 8nd to8 of the dam will lead to embankment failure. Conse- quently, overtopping is not permitted by contemporary design.

The PMF (probable maximum flood) has been used by th8 Bureau of Reclam8tion [l]’ as the IDF (inflow design flood) for new dam designs and for modification of existing dams when failure could cause potential loss of human life or significant property damage. Because of larger predicted storms using the current data bas8, ths PMS (probable maximum storm) and PMF magnitudes used for design of new dams and modification of existing dams have in- Cr88S8d significantly. By this criteria, many existing small embankment dams need increased spillway capacity or require greater dam height to eliminate overtopping. When designing new dams and spillways or modifying existing dams to meet revised restrictions--and often larger PMF-costs can become excessive. In some instances, it may not b8 physically nor economically feasible to accommodate large floods without overtopping. In fact, some embankment dams have been removed rather than undergo exorbitant costs of modification. However, some embankments have undergone moderate overtopping without failure [Z]. Therefore, it is surmised som8 existing embankment dams, especially those less than 50 feet (15.24 m) high, could possibly be modified to safely permit overtopping.

CONCLUSIONS Conclusion Based on Model Tests The conclusions here 8re based upon observations of single tests. At the current state of technology,

l Numberinbladmsmfertol#KJwlioenphy.

conclusions should be considered relative to each other; i.e., the model-aside from lack of repeated tests-is expected only to determine which of the treatments testted worked better, but not actually how much better. These conclusions are also specific to the single soil tested. With these limits in mind, it is concluded th8t:

1. For all treatments tested, flow eventually transformed into chute and pool mode.

2. Chute and pool flow had less scouring power compared with the less-rough plane shear flow, which occurred earlier.

3. The smooth hard crest cap extending 10 feet (3.048 m) down the slope caused a deep scour hole just downstream of the cap.

4. The flow, aher roughening the sloping part of the cap with epoxied pea gravel-representing fixed cobble roughness, scoured out one-hatf the embankment volume compared to erosion with the smooth hard crest cap.

5. Riprap, with 8 simulated maximum size of 24 inches (810 mm), immediately fluidized 8nd W8Sh8d Wt.

6. The mesh-contained treatments tested showed no indication of failure.

7. Flow on the embankment slope of 4:1, protected with gabions, scpured five times the volume compared to the gabion protected 6: 1 slope.

8. A change of compaction from 95 to 102 percent standard Proctor density decreased volume of scour by one-hatf.

9. Scour increased about 40 percent by doubling the unit discharge from 40 to 87 (fF/s)/ft [3.7 16 to 8.082 (m3/s)/m].

10. A slight sag any place along a crest axis combined with a slowly rising hydrograph c8n cause gully-type erosion and eventual breaching.

11. Flow, with approach head less than about one- twelfth of the crest length in the direction of the flow, will cause undulating flow on the crest; i.e., attempting to pass through critical depth more than once.

12. Design predictions 8re hampered by lack of verified 8d8qU8t8 governing equations for flow and sediment entrainment and transport for the steep shallow case.

13. Reservoir pool geometry can cause side and centrally located bottom eddies that will be inten- sified longitudinally by upstream embankment uplift into strong sediment transporting vortices downstream from the crest.

Conclusions Related to Modeling and Design Needs

The following conclusions are based on literature review and analyses. Some analyses are based on pro- posed equations and relationships that are not fully verified nor modified and/or tested for steep (tans greater than 0.08) and shallow (kJ4D greater than 0.1) flow. However, this analysis suggests possible research approaches. The conclusions also reflect design and modeling needs; they should help in the decision process for planning and monitoring possible future Bureau investigations or contract research.

1. Suitable flow and sediment transport equations need to be developed and verified to determine flow forces and subsequent erosion on embankments. Equations should adequately account for steep flow, shallow flow, and large bed form roughness relative to flow depth. For rock- protected embankments, the effects of large-scale roughness, element shape, and distribution need to be included. The equations should include the effects the rapidly accelerating part of overtopping flow.

2. Uniform flow equations do not apply to steep shallow flow or rapidly accelerating flow. Thus, for ovenopping flow, friction factors need to be determined from hydraulic measurements and more comprehensive equations as noted in conclusion 1. For shallow flow, with relatively large-scale roughness elements, friction factors are a function Of:

l Froude number F, l shear velocity-element size Reynolds number

U.dlv, l a projected drag area to flow area ratio, and l a concentration function, 1, of roughness ele-

ment spacing and distribution.

3. The separation of the energy slope, S, into the fractional part caused by form drag losses and the part caused by surface drag has been applied with some success in the bed form flow mechanics. This approach allows the similar separation of all hydraulic parameters such as flow velocity K hydraulic radius R,,, tractive shear on the flow boundary, r, and shear velocity U. . This approach may help in handling large element roughness.

4. Because overtopping flow makes transition from plane shear flow to chute and pool flow, the

steep flow friction factor, f, will vary with time as scour progresses. Different flow equations may be required during various phases of transition. To modify equations for handling chute and pool flow, will be difficult if at all possible.

5. For overtopping flow, model scaling sediment requires verified sediment transport equations that accwnt for soil types ranging from noncohesive to fully cohesive. Based on existing uniform flow equations, it is expected that fully cohesive soil transport can be scaled according to model length ratio and noncohesive transport according to length ratio to the 312 power. A transition scaling function likely exists between the fully cohesive to noncohesive transport of soils.

6. For a highly transient event such as a breach ing dam failure, model to prototype adjustment and verification is impossible. The difficulties are that model volumes and scour times are small.

7. Developing satisfactory equations for sedii rnent transport in steep and very rough chute, and pool type flow may or may not be possible but will require much research. The development of reliible design methods may actually require full scale laboratory and prototype field testing. Much long-term effort is needed to develop adequate governing equations for rate of sediment transport over steep slopes. In real&y, equations ara needed not only for sediment, time, velocity scaling, and for mathematical modeling, but needed for more rational inferences from one prototype experience to another and to new design cases.

8. Random aspects related to overtopping flow such as the many variables, jack of true soil hom- ogeneity, different soil classifications, and hydrograph variations present a strong case for more repetition of model tests and uniform documen- tation of failures in the field.

9. A well-coordinated interagency team approach is necessary to fund and make positive progress in solving a problem of this magnitude and expense.

10. The study of erosion during overtopping flow is a multivariable and multidisciplinary problem.

TECHNICAL CONSIDERATIONS AND ANALYSES

Geneml Scour Considemtions

Scour is a complicated interplay between soil properties, soil conditions, bed form, and flow characteristics. Furthermore, this interaction is complicated

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by the loose interface between the flowing water and the sediment bed. Flow can change bed form due to scour, dune movement, and deposits. A change in bed shape, in turn, changes flow characteristics. Moving large particles can abrade an embankment. When these larger particles are not moving, they can cause local scour in their lee.

Soils have diierent grain size, d, distributions which help identify soil types. Fine-grained soil identification is complicated further by variations of cohesiveness that can be partially expressed by the PI (plasticity index) and the LL (liquid limit). lt is desirable, but difficult, to express erosion resistance and transport in terms of soil properties, state of consolidation, construction methods, aging, and weathering. Ions in soil pores and flowing water can reduce or increase electrochemical surface activity bonding that produces cohesiveness.

Fluid shear, lii, drag, secondary flow, and turbulence are considered the main hydraulic factors affecting sediment transport. These factors vary with channel geometry, relative roughness, and Reynolds number.

Turbulence

A certain turbulent intensity may initiate movement. Once sediment is suspended, a somewhat less intense turbulence will keep particles in suspension. Turbulent intensity, s, , is the root mean square of the velocity fluctuations divided by the mean velocity; i.e..

where V’ is velocity fluctuation about V the mean velocity.

Values of E, have been measured from 0.03 to 0.07. However, 0.1 is considered the value at which the velocity fluctuation can no Longer be considered part of the main flow. Steep flow transition from plane shear flow to chute and pool flow is most likely related to this turbulent intensity near the 0.1 value. Turbulence at any point is strongly affected by flow section geometry, bed form, friction factor, f, fluid properties, location with respect to boundary and form disturbances just upstream.

Uniform Flow Equations for Tractive Shear Caused by Flow

For flow in canals and rivers, the simplest expression for average tractive shear, 7, can be determined from a free body diagram for normal flow.

5= y&S (1)

where: r = average tractive shear on the flow boundary

Rh = hydraulic radius = A/P A= flow section area P = flow section wetted perimeter S = slope of bed for normal flow or energy gra-

dient for gradually varied flow having short reaches

Y = specific weight of water = density p times sravity9

Friction head loss for open channel flow can be expressed by the Darcy-Weisbach relationship as h,:

(2)

where: h, = friction loss in water head

f = Darcy-Weisbach friction factor

= @(!!p)

L = reach length R,, = hydraulic radius V = average flow velocity g = gravitational constant (acceleration) v = kinematic viscosity

k, = rugosity, boundary surface roughness

Slope is the same as h,/L and specific weight, y, equals density, p, times gravity g. Thus, equations (1) and (2) combine:

fYF 7 = - = fpV2/0

89

This equation clearly shows the relationship of tractive shear and the Darcy-Weisbach friction factor and velocity. For turbulent flow, the interrelationship between Darcy-Weisbach friction factor, f: Chezy coefficient, C; and Manning’s, n, is:

1.49 Rh’fd

nfi (4)

Various investigators [3 and 41 and others, have related tractive shear to vertical velocity profiles by logarithmic relationships. These equations can be reduced to two-point relationships. One example is the following:

iv2 - w 7’

2.5 lW.(Y2/YO (5)

The two velocities should be measured close to the bed or for relatively small values of distance, Y, from the bed not so close to the boundary that the pitot tube or other measuring devices are affected by boundary proximity.

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Smeardon and Beasley [3] derived an equation for tractive shear for gradually varied flow using the definition on figure 1. The resulting equations for both velocity and unit discharge, q, are:

1 (6)

dD dZ 7 = yD (@/go” - l)--- + - 1 dX dX

(7)

For accelerating flow (dD/dX) is negative; for decel- leration it is positive. Comparison of these with equation (1) shows the complications caused by gradually varying flow. Note that equations (6) and (7) reduce to equation (1) for uniform flow and small slopes.

A momentum approach was used to provide more insight into the nature of this type of flow because of the steepness and high acceleration of overtopping flow. Using the definition on figure 2-where Q, is the slope angle between the bed and horizontal-the resulting equations for the velocity and unit discharge are:

e VdV dD ij .

‘5= --- gdx cosq, z + E =“v 1 (8)

dD B cosq, - + - sing,

dX D I (9)

For uniform flow and small slope, equations 8 and 9 also reduce to equation (1). Equations 8 and 9 govern design hydraulics and model flow over embankments and define, 7, for steep accelerating flow. Thus, equations such as (1) through (7) should not be used to input threshold tractive shear, 7,, values for com- puting any of the other variables using the steep flow equations. Steep flow friction factor, fs , and 7, are strictly dependent variables best determined by measurements with steep flow and using steep flow equations (8) or (9). For steep fully developed smooth nonaccelerating flow, equation (1) can be used by using sinp for the slope, S.

Limits of Uniform Flow Equation

Stable channel design and riprap design methods generally use uniform flow equations to obtain a design velocity or shear value and then use some form of particle entrainment function or critical velocity or shear value versus diameter. However, the uniform flow equations and friction factor function curves should not be used for relative roughness for kJ4R, greater than 0.1 where R,, is the hydraulic radius and k. is rugosity. Kamphius [5], and Brown and Chu [6] show that k, is about two times & , where & is the size of the bed of which 90 percent of grains are finer. Since overtopping flow is highly accelerating and rapidly becomes shallow, the kJ4R,, limit needs to be carefully considered for design and modeling.

Fllum 1. - DeMtion for gradually accelerating flow.

zF = f’QAV

V=- ;

dv,zL dD dx D’ dx

Fii2.- Dafinitimfor mamamum analysis.

Large Scale Roughness

Uniform flow friction loss equations have been used for rough, steep mountain flow. Also, soma efforts have been made to extend friction loss concept to large-scale roughness distribution and element shape characteristics. Although these approaches still use equations such as equation (2), the steep flow friction factor, f, , is no longer a simple function of Reynolds number and relative roughness. When relative rough ness is greater than 0.1, roughness elements pro- duce disturbances that reach the free surface causing gravity waves. Thus, shape and position distribution of the roughness elements become important. All equations and functions presented so far are limited by this maximum relative roughness value.

For example, Bathurst, Li, and Simons [7] propose the following relationship:

VL $=@(R.J x @(F) x @(A) x 9 $ =-

( 1 (10)

s 96s

where: f, = steep flow and large element roughness fric-

tion factor @ = function operator

5 = a veloclty-alement size Reynolds number = Froude number

A- roughness concentration parameter, a function of spacing, distribution of roughness elements, and element size relative to depth

A, = wetted area that forces flow between roughness elements

w= D= v=

= tf = g=

width of rough element flow section depth of flow section to the roughness datum average flow velocity gravitational constant (acceleration) hydraulic radius, A/P slope, horizontal to vertical

Friction values, f , determined by using functional relationships such as (lo), if fully defined, could be used in uniform equations such as equations (2) and (3) to

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calculate T values in equations (8) and (9). The opposite is true in that r values from equations (8) and (9) can be used in equations (2) and (3) to determine fs values, that can be used to determine functional relationship (10) to define steep flow after flow acceleration has been completed.

Secondary Flow

When flow approaching a dam is deflected upward and accelerates, relatively mild secondary flow and bank eddies are stretched out horizontally and intensify into strong vortices. The reservoir approach flow has a strong influence on the strength and location of these vortices. Friction resistance and the reservoir approach side and bottom geometry govern the approach secondary flow and eddies that are inten- sified by upward deflection. Vortices are strong sediment carriers. In fact, vortex action is often deliberately produced to increase sediment sluicing efficiency at diversion dams. Side entrance contrac- tions on the crest also cause vortex intensification. Thus, there are generally stronger vortices at each side of the crest with intermediate vortices across the remaining crest.

Even in symmetrical prismatic flow without upward deflected flow, secondary cell flow patterns develop. Knight and Pate1 [8] studied the structure of secondary flow in rectangular sections in terms of aspect ratio. Secondary flow cells will distribute along the boundary surface each with opposite rotation relative to its immediate neighbor. These investigators found that there is a step function, shown on figure 3, of aspect ratio versus the number of cells in one-half the flow section. Between opposing cell pairs, where circulation is downward, there is increased shear on the fiow boundary. When flow conditions are such that depth to width ratios are near the rising parts of the curve, additional pairs of cells can attempt to form or temporarily make additional pairs and then break up. This type action can cause increased scour.

Crest Hydraulics and Crest Length

Flow over the crest of an embankment dam is similar to flow over a broad-crested weir having a sloping approach. Bos (91 summarizes flow regimes in terms of H/L for a rectangular weir profile, where His total reservoir head above the crest and L is crest length (reach) in direction of flow. Understanding these regimes is required for insight into crest hydraulics. When

H/L < 0.08 , (11)

flow is subcritical over about 0.9 of the crest length (King [lo]); friction of the crest controls and undulations can occur on the crest. These undulations

I . l

L I I I I I I

0 I 2 3 4 5 6

ONE - HALF WIDTH + DEPTH

Fii3.- -ryflowcallsbrflowMction,Knight and Patal [8].

could increase scour and decrease the time for development of chute and pool-type flow. When

0.08 < H/L < 0.33 , (12)

smoother parallel flow exists on more of the crest and the coefficient of discharge is constant in this rangeofH/L.OnlywhilefIowmaintainsinequality(12) _ doestntesmoothbroadcrestedftowexist. When

0.33 c H/L c from about 1.5 to 1.8 , (13) -

parallel flow does not occur over the crest. Flow cur- vature causes increase in the coefficient of discharge,

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and control is near the leading edge of the crest over a separation cavity. When

H/L > about 1.5 , (14)

flow becomes unstable and, depending on comer sharpness, can spring free. At H/L of 3 or greater, the flow acts like sharp-crested weir flow and is stable.

Basically, critical depth controls flow upstream on the crest. Relation for discharge 0 at critical depth is:

(15)

lf the limits of equation (12) are maintained, the critical depth generally occurs somewhere in the downstream one-third to one-fourth of the crest for any shape channel.

where: Q = discharge

A, = flow area for the entire critical depth or control location which varies with discharge

g = gravitational constant (acceleration) a,= velocity distribution coefficient or kinetic en-

ergy correction factor v,” AA, / Pfic at the control section

T, = top width at the control section v, = velocity for an incremental flow area, AA, v, = average velocity of the entire control section

Using the equation of continuity; i.e., 0 = VA , squar- ing and dividing both sides by 2, equation (15) can be rearranged:

GV,’ A, -=- 29 2T,

(16)

Then specific energy, H,, at the critical location can be written as:

Hc = Q + PWT,) (17)

At an upstream or reservoir reference location, spe- cifkz energy, H,, can be expressed as:

Hl V12

= D, + a, - 29

(18)

For a significant friction loss, h,, the energy balance relative to crest elevation is:

H, = H, + h, (19)

and from equations (16) and (17):

DC = D, + a,V,2/2g - A,/2T, - h, (20)

Equations (15) and (20) define the crest hydraulics on an embankment. Values of 1.00 for a, and a, are probably adequate for design-with present knowl- edge. However, for water measurement and mathematical models, studies concerning the effects of crest end conditions and the effects of being near critical depth for the value of a need to be further researched and measured.

Boundary Roughness Changes

When a change from fixed bed to movable bed flow, or the reverse occurs, local scour is caused by slight boundary offsets and boundary layer roughness. As discussed by Schlicting [ 111, Jacobs [ 121 studied the development of vertical shear distribution after the change of roughness, for both the smooth to rough and rough to smooth cases. Jacobs study shows that for fully developed approach flow, the shear stress at the bed immediately takes the new value, . which is equal to that for fully developed flow for the new roughness. After a change in roughness, turbulence has to change its scale to adjust-affecting scour considerably.

Early Threshold Tractive Shear and Velocity --pts

To attain useful scour and transport criteria, soil properties must be related to flow properties. For cohe- sionless soils, it is generally accepted that the mass of the largest sediment grain transponed is propor- tionally related to velocity, V, raised to the sixth power. This proportionality suggests the possibility of a threshold of movement velocity, V, , that will just move a particle of diameter, d. tf particles are assumed to be spheres with constant specific weight, then weight, W, is proportional to the third power of diameter. Combining these two proportionalities results in:

V, a d’” (21)

Also, equation (3) combined with equation (20) suggests the possibilhy of a threshold traotive shear, r, , or shear that will just move a particle of diamter, d, and:

7, a d (22)

Relationships such as equations (2 1) and (22) are fre- quently used with some success. These relationships are really oversimpliications in terms of soil properties, soil conditions, and hydraulic conditions.

Generally, it is accepted that cohesion plays the most important part in scour resistance of clay soils. Sometimes the effects of cohesion are expressed by assuming that they are defined by a single soil parameter such as grain size only. Sometimes more complicated approaches assume that cohesion, C, ,

7

is a function of different combinations of soil parameters such as:

& = mean grain size, o = standard deviation

grain size,

C, = compressive strength,

Kd = skewness of grain size, PI- plasticity index,

%C = percent clay, LL = liquid limit, e=vokisratio,and

S, = vane shear strength, others.

Rebtive Scour Resistance in Terms of Soil Classi& catimlzlnd~

Throughexperience~engineersdevelopCWCt3ptSOfr& ativemsistanceofsoilsagainsteroskm intetmsofsoil . . V Forexample,Burwu0fRedam9ti0ns0ik eng&rem[l3]hsverankedthere&tivem&tance .- ferentsoiltypesinmble1.TNsrankingisfor~ where soils were taken from borrow areas and recompacted-

When soils have 50 percent of their grain particles smaller than 0.074 millimeter in diameter, then plastic properties can contribute to erosion resistance in varying degrees from just slightly adding to the effects of grain size to being the dominant source of resistance. Plastic soil properties can be expressed at least partially by the following Atterberg limits [from ASTM (American Society for Testing Materials)]:

Liquid Limit (LL)-The moisture content corre- sponding to the arbitrary limit between the liquid and plastic states of consistency of a soil.

Plasticity Index (PI)-Numerical difference between the liquid limit and the plastic limit.

Plastic Limit (PL)-The moisture content corre- sponding to an arbiiafy limit between the plastic and the semisolid states of consistency of a soil.

Figure 4 taken from Gibbs [13] shows the relatiie erosion resistance of cohesive soils in terms of Lf and P/. The “A-line”’ separates the clays from the silts below. lt should be remembered that the thresh old tractive shear ranking in this chart applies to dis- turbed soils that were recompacted to 90 lb/h3 (11.34 N/d). The threshold tractive shear ranges of the laboratory and field data used to derive this value are given next to the shading key (fig. 4). The plot indicates that the highest cohesive soils occur in a zone around Lf of 35 percent and a PI of 15 percent.

Smeardon and Beasly [3] did correlations of threshold tractive shear with each of diierent soil properties, that included Pi, dispersion ratio, percent clay, and

‘Eiwth Mimal, %weau of Reclamation, 2d ad., reprint 1995.

mean particle size. Df these variables, they selected the PI and dispersion ratio as more strongly correlated. However, mean particle size correlated about as well. This is probably the result of electrochemical surface activity being strongly related to particle size. Data for these correlations were obtained in a flume with one clay reformed for tests.

Carlson and Enger [ 141 did multiple correlation analyses of various combinations of soil properties versus threshold tractive shear values with reformed samples of clay. They found that plastic properties and densities are the most important soil properties that affect scour resistance.

Tables 2, 3, and 4 were taken from Lane [ 151 who summarized the work of early investigators concerning threshold velocity and tractive shear. These tables include a few data for cohesive soils.

Table P-Etchevery’s data for alluvial and clay soils ranged from about 0.10 to 0.43 lb/h2 (4.8 to 20.8 Pa). Table 3-Fortier and Scobey’s data for alluvial silts to stiff clays, ranged from about 0.05 to 0.25 lb/ft2 (2.4 to 12 Pa) for clear water and 0.15 to 0.48 Ib/ft3 (7.2 to 22 Pa) for water transporting colloidal silts. Table 4 data ranged from about 0.02 to 0.63 lb/ft2 (1 to 30.2 Pa).

It should be noted that these ranges include values of threshold tractive shear much higher than those obtained by Gibbs, Carlson and Enger, and Smear- don and Beasly for recompacted samples.

Kelly and Gularte [ 161 developed a threshold tractive shear equation that accounted for salt ion effects upon the cohesive strength of soil and a standard base measurement of rupture shear force angle.

The Bureau did threshold tractive shear tests on bed- ded clay samples taken from downstream of the Grand Coulee Third Powerplant. Several modes of erosion such as flaking and chunking were noted. Threshold tractive shear values ranged from 0.06 to 0.27 lb/n2 (2.87 to 12.93 Pa). Most of the observed modes of erosion were related to segregation of grain sizes and weaker cohesion in thin layers. Some segregation often occurs during construction of embankments.

Kamphius [17] determined that threshold tractive shear values for cohesive soil vary with:

compressive strength, vane shear strength, plasticity index, percent clay, and consolidation pressure.

.

8

Table 1. - Soil ckwsification with relative erosion stabiiii

Majordiviskmofsoils Typical names of soil groups Group sYnlbols

Erosion resistance*

Pine-grained aoils~

Silts and cbys LL greater than 50

Inorganic silt, micaceous or diatomaceous fine sandy of silty soils, elastic silts

Inorganic clays of high plasticity, fat clays

Organic clays of medium to high plasticity

Inorganic silts and very fine sands, rock Rour, silty or clayey fine sands with slight plasticity

MH

CH 12

OH

ML

Silts and clays LL less than 50 Inorganic clays of low to medium plasticity, gravelly clays,

sandy clays, silty clays, lean clays CL 11

Organic silts and organic silt-clays of low plasticity OL

Coaree-gmined soil3 saniw Siity sands, poorly graded sand-silt mixtures SM 10

sands with finas Clayey sands, poorly graded sand-clay mixtures SC 7 (appreciable amount of fines) Sand with clay binder SW-SC 6

Clean sands Well graded sands, gravelly sands, lie or no fines SW 8 (lie or no fines) Poorly graded sands, gravelly sands, lii or no fines SP 9

Grard@ Silty gravels, poorly graded gravel-sand-silt mixtures GW 5

Gravels with fims (8ppre&ble amount Clayey gravels, poorly graded gravei-sand-clay mixtures GC 4 of fines)

Gravel with sand-clay binder GW-GC 1

Clean gravels (liie or no fines)

Well graded gravels, gravel-sand mixtures, lii or no fines

Poorly graded gravels, gravel-sand mixtures, llle or no fines

GW

GP

2

3

Worethsnons-hslfofcosrsefmctionisbrgsrthsnNo.4sisvesize. ‘Numbers it&We ths order of increasing vskms for ths physics1 pmpsrty nsms. Numbers indicate rebtivs suitability (1 = besti.

Entrainment Functions

Simons and Senturk [ 181 showed by dimensional analysis that noncohesive material has an entrainment function expressed as:

ft =

d= s

*: =

71 =Q U.,d d A -,-a- St = @. - PI mf V %, P )

(23)

v, = where: s, t, al w = subscripts for sediment, threshold and 7, =

water

friction factor at threshold of sediment motion

sediment diameter gravitational constant (acceleration) hydraulic radius at threshold of sedi-

ment motion slope of energy gradient at threshold of

sediment motion velocity at threshold of sediment

motion tractive shear at threshold of sediment

motion

I EXPLANATION

Hi qhcst resistance to erosion 0.045

S I ig ht erosion expected 0.035

El Moderate erosion expected

0 025 . X I.. I oeo,5 z<: l-l Low resistance to erosion

SUGGESTED UAJOA DIVISION I

0 0 IO 20 30 40 50 60 70

LIOUID LIMIT

Fiium 4. - Erosion chsmcte+tics for ke-gmined cohesive soils with respect to plasticity, GUS (131.

= density of water $ = function operator v = kinematic viscosity

u 01 = threshold shear velocity = V, m = vm = v* and is a measure of turbulent intensity

This function has been defined empirically by Shields [ 191 after neglecting d/R,,, for relatively deep flow and p,/p as relatively constant for sediment and water. Shields’ entrainment function is shown on figure 5.

Using the pi term, d/R,,, , and valid pi term manipulation, d can be optionally replaced by R,,, in any of the other pi terms resulting in an equally valid but undefined relationship.

rt =@ U&t d P.

@. - PI9 Rht -, -* - (24)

V &t P

This manipulation reduces the use of d which is cum- bersome to determine for clays and increases the use of hydraulic radius which is easier to establish.

The pi term, qtl[R,,,(p, - pig], is considered a dimensionless shear or a shear velocity Froude number squared and the term, U., R&/v, is the shear velocity flow section Reynolds number.

For clay, dimensionless cohesion, C, ; and density or compaction, PC ; parameters must be included. Again dropping the last two pi terms because of relatively deep flow, or relatively fine sediment, and for water and constant 7, :

rt =Q U&t @. - PI 9 RM

- , c, , P, V 1

(25)

This equation shows that threshold shear, rt , determined from a test device should be used for design with caution and an adjustment really needs to be made to account for lack of complete hydraulic similitude between the test device and actual flow. Krishnamurthy [20] gave an example with a comparison of threshold tractive shear values obtained for clays by a jet and a shear flume flow device. The threshold values were from 10 to 50 times greater

10

.

Table 2. - Comparison of Etchevery’s maximum allowable velociiies and trsctive forces-from Lane [ 15).

Manning’s VW

used, n

.025 6.00 to 8.00

0.75 to 1.00

1.00 to 1.50

1.50 to 2.00

2.00 to 2.50

2.50 to 2.75 2.75 to 3.00

3.00 to 3.75

4.00 to 5.00

5.00 to 6.00

0.006 to 0.001

.Oll to .025

,025 to .045

.045 to .070 .070 to .os4

.084 to .lOO

.lOOto .157

.278 to A34

.627 to .903

.627 to 1 .114

1 ft/s = 0.3048 m/s, and 1 lb/W = 47.88 pa.

for the jet device relative to those obtained with shear flow device. Thus, if the function in equation (25) was defined, it might help to evaluate test facilities and to account for scale effects between erosion test facilities, models and flow channels with cohesive beds.

Gravity Effects on Erosion Resistance

Because riprap and gravel blankets could possibly be used to protect embankments, slope-gravity effects were considered. The assumptions of Carlson [21] were used. The main hypothesis used was that resistance to motion on a side slope and on the flat bottom is equal to the normal force times the tangent of the angle of repose, 0, of the bed material. Angle of repose, 0, is the angle between the horizontal and the maximum slope that a soil assumes through natural processes. For dry granular soils, the effect of the height of slope is negligible; for cohesive soils, the effect of height of slope is so great that the angle of repose is meaningless (ASTM). Brooks [22] ex- panded the relationship for flow on the slope in any direction resulting in:

~JG = e/v: (29) Equation (26) is derived by setting the ratio of re- sultant forces tending to cause motion to the forces resisting motion equal to the tangent of internal shear resistance angle. For noncohesive soils in loose state, angle of repose, 8, is nearly equal to the shear resistance angle just at rupture. Figure 6 indicates that noncohesive material with angle of repose at 34 degrees on 6: 1 downhill slope has about 75 percent of the critical tractive shear resistance of that for the same material on a flat bed. As this slope increases, erosion resistance decreases rapidly; as slope de creases, erosion resistance increases slowly. Non- cohesive material on a 4:l slope has about 60 percent erosion resistance relative to flat bed flow resistance. For cohesive soil, equations (27) through (29) need to be modified by incorporating Coulomb’s equation for shear resistance at rupture.

(26) Sediment Transport Functions

When /3 is 0 or when flow parallels the canal side slope in the direction of the channel axis, the equation simplifies to Carlson’s previous relationship.

b

;= cosO( 1 - s)ln=#l -(z)’ (27)

where: T* = threshold tractive shear for a sediment par-

ticle on a slope z, = threshold tractive shear for same particle on

a level bottom 8 = angle of repose for noncohesive material B= angle of flow relative to a horizontal line in

the plane of the embankment face v= slope, angle of canal or dam embankment

s&l= subscripts for a sloped bed and a level bed

For this study, an analysis was made for uphill and downhill flow resulting in:

fs sin(0 f 9) -= (28) 5 sine

lt can be shown that Brooks’ general equation (26) reduces to equation (28) for /j of 90 and 270 degrees. This function is plotted for both the uphill and downhill case on figure 6.

Angle of repose, 0, varies from 27 to 42 degrees. Figure 7, taken from Simons [23], shows that angle of repose varies with size and angularity of particle. For convenience in riprap design, threshold tractive shear can be related to velocity using equation (3) which results in:

Shields’ entrainment function, figure 5, can be considered a transport function for a special case where

11

Table 3. - Comparison of Forth and Scobey’s limiting velocities and trective force values-from Lane [ 15).

Material

Fine sand colloidal Sandy loam noncolloidal Sift loam noncolloidal Alluvial silts noncolloidal Ordinary firm loam Volcanic ash Stiff clay colloidal very Alluvial silts colloidal Shales and hardpans Fine gravel Graded loam to cobbles when noncolloidal Graded silts to cobbles when colloidal Coarse gravel noncolloidal Cobbles and shingles

1 ft/s = 0.3048 m/s, and 1 Ib/ft2 = 47.88 Pa. n = Manning’s n

R

0.020 .020 .020 .020 .020 -020 .025 .025 .025 .020 .030 .030 .025 .035

For clear water

Velocity Tractice force fib Ib/ft2

1.50 0.027 1.75 .037 2.00 -048

%-kit .048

2:50 .075 .075

3.75 .26 3.75 .26 E .67

-075 3:75 .38 4.00 .43 4.00 .30 5.00 .91

Water transporting colloidal silts

Velocity Tractive force WS Ib/ftz

2.50 0.075 2.50 .075

:-it .ll

3:50 .15 .15

3.50 .15

55:E .46 .46

58:E .67 .32 .66 5”E .80

6:OO .67 5.50 1.10

Table 4. - U.S.S.R. limiting velocities and tractive forces in cohesive material-from brie 1151.

Compactness of sediment bed Descriptive term Loose Fairly compact Voids ratio range 2.0 to 1.2

Very compact Compact 1.2 to 0.6 0.6 to 0.3 0.3 to 0.2

Principal cohesive Limiting mean velocity ft/s and limiting tractive force Ib/ft2 material of bed ft/S Ib/ftz W lb/*’ W lbfft2 W Ib/ft’

sandvdaysl~content less than 50%) : 1.46 0.040 2.95 0.157 4.26 0.327 0.630

Heavy clayey soils : 1.31 .031 2.79 .141 4.10 .305 ~*~ .563 -YS : 1.15 .024 2.62 .124 3.94 .281 5:41 .530 Lean c&yey soils : 1.05 .020 2.30 .096 3.44 .214 4.43 .354

1 h/s = 0.3048 m/s, end 1 lb/ft2 = 47.88 Pa.

transport is nearly equal to zero. Gessler’s [24] modification of Shields’ function, figure 8, with parallel cuwes for diierent probabilities of moving, further indicates that transport is related to the Shields’ parameters and approximately parallels Shields’ critical curve. Graf and Pazis [25] make an even firmer con- nection to transport by developing nearly parallel curves, figure 9, to Shields’ curve in the transition zone in terms of the number of particles moving. Thus, it seems quite logical that Shields’ parameters should define higher and very active transport as well as incipient motion of sediment. For modeling, the most useful form for noncohesive sediment transport function is given by Taylor, figure 10, and discussed by Vanoni [26]. Taylor uses a dimensionless sediment discharge parameter as a third or nesting parameter along with Shields’ parameters (fig. 5). The dimensionless sediment discharge parameter, 9,. , is defined as:

9r = 9clU.d (30) where:

9. = sediment discharge in volume per unit width per second

u. = shear velocity d = diameter of sediment particle

Taylor curves of dimensionless discharge, 9,. , are approximately parallel to Shields’ curve, including the transition dip, figure 10. The extension of this function for larger sediment discharge would be of con- siderable benefit to modelers. Thus, model sediment discharge scaling requires investigating the following functional relationship:

9‘ U,d= 9.. (31)

Replacing d with Rfi by pi term manipulation and adding dimensionless cohesion and compaction terms might be a good approach for investigating cohesive transport. Then the undefined but equally valid transport function can be written as:

Q.=@ 7 U.4, U.Rll (j)c-p)gR*’ y-# co. pc (32)

.

12

Svm I Description 1

Amber Lignite (Shields) Gronite Barite Sond (Casey) Sand (Kramer) Sond ( U.S.W.E.S.) Sand (Gilbert) Sand ( Vanon i ) Sand (Gessler)

Ys, g/cm3

1.06 1.27 2.7 4.25 2.65 2.65 2.65 2.65 2.65 2.65

I I I Ill11

Motion

t

t iym

7 A *

No motion

Sond (Casey) Sond (Kromcrl Sand ( U.S.W.E.S.1 Sand (Gllbcrtl Sand ( Vononl)

0.001 I I I l11111 I I IIIIIII I I llIlrl I I1111

I I lllll

I.0 IO 100 loo0

Figure 5. - shields [ 1 S] diagram for threhold of bed material movement. .

13

v, = V” =

Velocity at threshold of sediment motion for flow over level bed Velocity at threshold of sediment motion for flow over sloped bed

2 = Horizontal component of bed slope r,, = Tractive shear at threshold of sediment motion for flow over level bed 7 z

= Tractive shear at threshold of sediment motion for flow over sloped bed = Angle of repose for bank material

Q, = Slope angle of embankment

Flat

Gs ViL -a- % V2 dl

0.00 38 40 42

ANGLE OF REPOSE, 8

Z-l.25

Figure 6. - Embankment stabilii correction factor for slope grmrity effect during uphill and downhill flow.

14

: w ‘. = 2

32

30

I I I

I 0.01 .02 Do3 Al xl6 .oll 0.1 .2 .3 .4 .6 .I) I.0 2 34 6 s IO 20 3040 60

DIAMETER OF MATERIAL. INCHES

Figure 7. - Angle of repose for rock materials, Simons (23).

0. IO

0.08

0.06

' ' QGesslers data l Shields data

60 80 100

R*

“Sd =-

U

Figure 8. - Gessier’s [24] probabilii rnodiicetion of Shields’ [ 1 g] function.

15

0.11 0.0 0.0

0.0

0.0

= O.O!

b I: '; P 00 . 4

II * 0.0;

h

0.02

5.

I-

5-

I,

IO loo 1000

R* “ld v-

V

I I I I1111 I I I lllll I I Illlll I I

I Illlll I I II111 I I l l l l l

I 3 6 IO 30 60 too 300 600 IO00

RS f U*d Y

Figure 10. - Taylor’s dimensionless sdiment disdwrge plotted with Shields’ function, Vanoni [26].

16

For shallow flow, relative roughness (kJ4R) would have to be added. Further adjustment in terms of the parameters such as in equation (10) also would be needed to account for large scale bed form roughness.

Metha et. al. [27] recommended Kandiah’s [28] equation that indicates mass erosion rate varies linearly with excess shear (r - ?J and expressed erosion rate as:

where: 9m =

‘5= 7, =

k,,, =

(33)

mass of sediment per unit area per second tractive shear on the flow boundary threshold tractive shear an erodibilii constant

Chen [29] uses a volume form of excess shear equation developed by Ariathurai and Arulanadan [30] expressed as:

9” = kv(r- 78 (34)

where: 9” = volumetric sediment discharge for unit area

in (ft3/s)/ftz k, = 0.00005 for both cohesive and noncohesive

soils n = 1 .O for cohesive soil and 1.5 for noncohesive

soil

General Model Similitude

To represent actual conditions, a model must be gee- metrically, kinematically, and dynamically similar to the prototype. lf a model is not fully similar in any one or more of these three aspects, care must be taken to account for or minimize distortion. At times, models ‘can be physically distorted in ways of less importance tending to compensate for the defect of other more important variables. Occasionally, with care, data from a defective model can be corrected by analysis or previous experience. However, if distortions are too large or too numerous, a model will not represent the prototype in any way and inter- pretations or data correction by analysis will be fruii- less. These same similitude considerations also apply to combining results from diierent shear erosion test devices or to applying a threshold tractive shear value from any one test device to an actual river or channel case.

Hydraulic Friction and Flow Scaling

The most accurate and efficient modeling can be ac- complished by using the similiide and approximation method of Klein [31] when governing equations are available and applicable to both the model and and prototype. For model analysis, the equations are

put into dimensionless form by normalizing the variables by defining dimensionless variables using characteristic boundary values to replace dimensional variables. Characteristic or boundary values and physical constants are separated from the dimensionless variables into groups in each equation term. Dividing one of the groups into all the remaining groups results in dimensionless parameters or (pi) terms which are attached as coefficients to specific terms of the dimensionless equation. These pi terms can be evaluated for both the model and prototype. Thus, scaled effects of a model can be quantified provided the characteristic boundary values in the pi terms have been carefully selected.

For overtopping flow, using equation (8) and applying the similitude and approximation method of Kline [31] results in:

dD. B.

- ‘OS’ dX. - + -ii sing (35)

.

where: 7. = 8r/fpe v. = v/v, x. = x/xc

B. = B/x, D. = D/X=

dV. = dV/V, dX. = dX/X= dD. = do/x,

Xq = flow channel slope angle

VL = linear characteristic boundary value = characteristic boundary velocity value

Asterisks denote dimensionless variables, c denotes characteristic or bondary values. The term e/gX= (the Froude number squared) and f (the Darcy-Weis- bath friction factor) should be the same for both a model and its prototype. Satisfying this requirement would ensure that forces, turbulence, and secondary flows are similar between model and prototype.

Making the Froude number and friction factor the same for model and prototype is diicult to do and often cannot be achieved. However, effort should be made to determine by means of the dimensionless equation the degree and effects of any deficiency in the interpretation and use of data obtained with a model. By tradition, the model value for vf/gX, is made equal to the prototype for a free surface model making necessary use of analysis, field data, and best estimates to check friction, sediment entrainment, and transport scaling.

17

In equation (35) the pi term attached to shear term is a product blend of friction factor f and Froude number squared, F2. For Froude scaling, the Froude number is set equal for both the model and prototype. Thus, f should be made the same for both the model and prototype. The friction factor, t is a function of Reynolds number, 4R,, V/v , and relative roughness, kJ4R,, , and the modeler can check frictional scaling with Nikuradse-type friction curves.

In deriving equation (35) dimensionless shear 7. was defined as:

7. = 8r/pfe

Using equation (37), 7. can be redefined as:

Sediment Transport Rate Scaling Substituting into equation (35) results in:

Settling velocity or fall velocity is closely related to the phenomena of sediment being entrained in the flow and how long it will travel once lifted off the bed into the flow. Settling velocity scales by Froude law when model sediment diameters are equal to or greater than 1 .O millimeter.

lf equation (31) is accepted, then to scale noncohesive sediment discharge, the dimensionless sediment discharge parameter qJlJ.d should be made the same for both model and prototype. On this ba- sis, sediment discharge would scale as L pi2. 8agnold [32] shows that rate of noncohesive sediment transport is proportional to 7 to the 3/2 power times a soil erosion efficiency constant k. If the efficiency constant for model and prototype are equal, then by Froude law 7 scales according to L, , and sediment volume transport rate, q. , scales as L,‘12.

tf an excess tractive shear equation like (33) is accepted for cohesive soil and for dimensional units of k-same as for q., then

7- 7 Q=k+

* (36)

For field recompacted cohesive soils of canals 7t is generally less than 0.07 Ib/ft2 (3.31 Pa). For geolog- ically compacted clay 7, is generally less than 0.40 Ib/ft2 (19.15 Pa). Tractive shear such as would occur for steep Row is large, greater than 15 lb/h2 (718 Pa). Thus, I; can be ignored in the numerator and

(37)

Using equation (3) and (37) results in

f v2 a =kp-- (1 8 7,

18

For a given soil, (7,) in the denominator in these equations can be incorporated with the soil erosion constant k and cohesive sediment discharge would scale as L,, the length ratio. lt is noted that V2/7, is the same term that Abt [33] uses for the transpon function.

=-

dD. ii

cosp,dx. I) + -zi- sincp

(39)

(40)

Thus

In this equation, k has the same dimensions as q.. Also, k, f, and 7, must relate to modes of flow such as chutes and pool. In modeling, the Froude number squared e/gX, is made the same for both the model and prototype. When a model scale L, has been selected, Reynolds number is distorted making manipulation within functions such as equation (10) required to attain equal f for both model and prototype.

Based on equations (31) and (32) and complete similitude, embankment sediment discharge scales to Lfi2 for noncohesive soils possibly making transition toward a linear relationship with respect to L , as soils become more cohesive. However, cohesive sediment discharge rate and scour time scaling are really contingent upon completely satisfying equation (40) or by making adequate scale adjustments by means of the equation for any model distortions deliberately or unintentionally incorporated in a model.

For ideal scaling, there also should be transient time similitude in terms of roughness and mode of flow changes during scour changes of bed shape.

LABORATORY MODEL STUDIES

Model Study Program

To help resolve scaling problems and for design considerations. it was decided that three typical embankment soils would be tested. Also, because of

known and possible limits of the transport equations (29), (33), and (36) it was originally decided to use three diierent unit discharges for tests. Later, however, tests were mainly directed toward supporting Bureau of Reclamation designers and the National Park Service in modifying two dams in Blue Ridge Parkway, North Carolina, to permit some overtopping without major erosion. Thus one type of soil was tested.

Modal Scaling Used for Laboratory Test Runs

Scaling is limited by not having a fully verified cohesive transport equation. Equation (40) is fully valid on the right side. However, the left side is only as valid as equation (37) and the care in its use. The variation of f, possible variation of the erodibility constant, k, during erosion and changing modes of flow still preclude the use of equation (40) for exact scaling.

Without verified equations, fully reliable sediment transport time and velocin/ scaling relationships can not be determined. This is not only a problem in using a small model but is also a hindrance in making predictions from experience with one full-sized embankment to another full-sized embankment. To estimate sediment transport scaling without equations, modelers generally compare a nonrandom prototype event of significant sediment transport quantity and time duration with model performance. This is not possible with a highly transient model such as dam failure.

Froude scaling applies to the crest hydraulics where friction does not play a significant part. A length ratio, L, , of 15 was selected to make the 2.12-foot-high (646-mm) model dam represent a typical National Park Service embankment dam about 32 feet high (9.754 m). Thus, by Froude Law:

D,= 15 depth of flow ratio, prototype- model

: = 58.1 unit discharge ratio = 871

v: = total discharge ratio

3.87 velocity ratio T, = 3.87 hydraulic time ratio R, = 58.1 Reynolds number distortion ratio

Kamphius [5], also, Brown and Chu [6] show that rugosity, k., is two times the & size of the sediment bed material. Since & was 2 millimeters, in the model, the k. simulated by scaling was 60 millimeters. It is expected that the model is too rough and velocities tend to model low and depths high.

Because of the lack of a fully verified transport equations, lack of Froude scaling for shallow flow, the model results are considered qualitative, more likely

indicating which treatment of those tested worked better rather than determining actually how much better.

Laboratory Test Facility

A typical National Park Service embankment dam, Blue Ridge Parkway, was represented in the sectional model. The model represented a 32-foot-high (9.754-m) dam and was constructed of materials similar to those of the prototype dam. The model embankment overflow represented about 4 feet (1.219 m) of water overtopping the crest of the dam for a period of 4 to 6 hours. The test embankments were placed in a 3-foot wide by 4-foot deep by 3(1 foot long (9 14-mm by 12 1 g-mm by 9144-mm) flume shown on figure 11. The 3-foot flume width represented 45 feet (13.7 16 m) of crest length. The model dam was 2.12 feet (646 mm) high and the crest extended 1.58 feet (482 mm) in the direction of flow.

Water was supplied to the flume by a portable laboratory pump. Discharges were measured by an 8- inch (203-mm) orifice venturi meter. One side of the flume had windows for viewing erosion and flow action. A nominal 1 P-inch (300-mm) diameter pipe with a gate valve was used to pass flow around the test embankment. Upon closing the valve, the reservoir water level would rise and flow over the test embankment.

Soil Tested

Soil used for all test runs was a clayey sand found near Denver. The soil had 45 percent fines passing the U.S.A. Standard series No. 200 sieve and 6 percent gravel retained on the No. 4 sieve. The soil gradation curve, shown on figure 12, indicates that the d,, size is 19.1 millimeters and the $0 size is 1.9 millimeters. The soil has an K (liquid limit) of 25 percent and a PI (plasticity index) of 9 percent.

Soil was placed and compacted to the desired test density of 95 percent standard Proctor for all but one test. The placement for test run 8 was overcom- pacted to 102 percent. The moisture compaction- penetration resistance curves for the soil are plotted on figure 13. Soil compaction was controlled by determining the soil mass required to fill 3-inch (76-mm) thick horizontal layers of model embankment soil, and determining the amount of moisture that needed to be added to obtain optimum compaction. To obtain better bonding between layers, the top of each layer was scarified before placing the next layer.

A 6: 1 slope was used for early tests because many investigators intuitively believe that this is a rational erosion resistant slope. This was previously discussed in section “Gravity Effects on Erosion Re- sistance.” In later tests, 4:l slopes were used

19

/Bypass pipe r Bypass valve

Water supply pipe

Return Pump and Settling Basin

\-Soil embankment

SECTION A-A

because of the high stability experienced with tests with gabions.

ModeiOpcHation

For most of the nine test arrangements, the unit discharge represented was 40 (ft3/s)/ft (3.716 (nV/s)/m]. The arrangement for test run 7 was rep licated for test run 6 but operated at a unit discharge representing 67 (ft3/s)/ft [6.062 m2/s)/m]. The first test run fasted only 17 minutes because erosion was considered to be excessive and more consideration needed to be given to the boundary effects of the model and the smoothness of the hard cap at the crest. Therefore, 17 minutes (about l-hour prototype time) was used as a common model test run time interval for the remaining tests to compare after erosion measurements. Some test treatments were og erated an additional hour which would be representative of 5 hours total of overtopping which is expected for the National Park Service dams being considered for modiication. A surveyors level was used to determine bed profiles and cross sections. A prism-end area method was used to calculate erosion volumes.

Desoriptlon of Slope Treatments Tested

The following are descriptions of embankment treatments tested with run numbers related to schematic sketches in table 5.

Run 1: Hard crest cap.-The crest was protected with a hard cap shown in table 5. The cap extended 10 feet (3.05 m) down the slope and ended with 7-foot (2.13 m) vertical toe curtain. Soil was placed in the model at 95 percent maximum Proctor compaction. Simulated overtopping flow was 40 (W/s)/ft r3.716 (m3/s)/m].

Run 2: Fiied cobble roughness on downslope part of orest oap.-Crushed pea-sized gravel roughness was epoxied to the downhill slope of the crest cap. The pea gravel represented rock roughness from 3 to 6 inch (76 to 152 mm) size. The previous tested embankment was excavated beyond the erosion and the soil replaced at 95 percent maximum Proctor density.

Run 3: Downslope riprap.-A 30-foot (9.144 m) slope length of riprap was placed just

20

. . \ A

.

HYDROMETER ANALYSIS GRADATION TEST TIME RIADIWGS I

SIEVE ANALYSIS

2S hr 4s nln 13%

U.S. STANDARD SERIES I

loo 60 min I9 nin

I I

4 min I nin 1200 I 1108 130 140 130 116 a10 I8 I I I I I I 1 I I I , -4

d I I I I lllll II I II I I Ill I I I I i

4 e 4 Js 1: 1 I iiilrrtr I I lb

mm’ -1 r .OOl 302 .oos 309 .019

CINLS

CLASSIFICATION SYMBOL SC- CL Gravel 6 n

45 K Send - Fines ,-, ‘9 96

.0-n .oir J-49 .A7 -*- 40 r.;r Of P

I MEDIUM

ATU33BERQ LIMITS SPECIFIC GR Liquid Limit 25.0 % Minus k. 4 2.66 Pla8ticity Index 9.0 %

J 1111 I I I Ill t I

3 I.” I

2.29 4.h ERS

I GRAVEL ._- -_... 1 COARSE 1 flNL COARSE

COGGLES

&VITY

CLGAR SQUARE Of IWINGS

Figure 12. - Gredatkm test of soil used fur test runs 1 through 9.

COMPACTION- PENETRATION RESISTANCE CURVES

600

200

0

I25

100

5 10 I5 20 25 MOISTURE, PERCENT OF DRY WEIGHT

I i i iI

t i ii i

5 IO I5 20 25

MOISTURE, PERCENT OF DRY WEIGHT

r

,

Figure 13. - (lolmpev redstence alwes.

22

Table 5. - Results of overtopping flow model-summary.

sketch unit E*W Rsbtivs vohnns Msrimlm MSXimm 8nd tints erosion mm pof

M- 5. unknstlnt7 dsct. dlgzt. comprriron baleI. scour scour

ti lmurs locmian locstion

tv M t44) 6) I@

.

40 1 2.41 -7.5 10.0

15.2 15.2

40 1 1.09 0.0

10.1

9.6

106.2

1 2.03 -5.9

10.1

9.0

10.1

0.42

0.60

9.9 40.8 62

9.1 40.6

3.0 95.1

2.19

0.94

-1.3 3.4 52.6 59.2

5::: a.3

104.3

0.63

1.00

1.00

2.02

0.00

2.5 10.0 63.7 63.7

-0.3 6.0 0.0 77.3

40 5 -2.5 0.0 2:

-0.3 3.9 0.0 tos

-0.9 5.6 0.0 tos

67 1 1.42

mTE:lit~o.jorem tft/s-0.3046m/s

downstream of the cobble roughened sloping part of the crest cap. The model riprap represented about 6 to 24 inch (152 to 6 10 mm) prototype riprap. An appropriate filter base was placed underneath to prevent embankment material from leaching through the riprap.

Run 4: Gabions.-Gabions were placed on the 6:l slope; 30 feet (9.14 m) long starting at the end of the cobble roughened downstream sloping part of the crest cap. The upstream compartments were securely anchored to the vertical curtain wall of the cap with epoxy. Flexible hook rug backing was used to model the mesh. The model represented 3- by 3- by 3-foot (0.914- by 0.914by 0.914-m) gabion compartments. The gabions were further anchored to buried timbers in the embankment.

Run 5: Gabiins with slope incmased.-Be- cause of the success in the previous test, it was decided to try gabions on a 4:l slope.

Run 6: Rock mattress .-An 18-inch-thick (457- mm) mesh contained rock mattress was anchored to tee-shaped cutoff wall at the upstream corner of the flat part of the crest, covering the flat part of the crest and extending 50 feet (15.24 m) down the downstream 4: 1 sloping face of the bank. The vertical leg of the tee extended 6 feet (1.83 m) below the flow boundary. The buried base of the tee was 5 feet (1.52 m) long. For this run only, the model soil on the 3:l sloped approach was protected with an 18-&h layer of g-inch maximum and 4-inch mean diameter riprap (457-, 229- and 102-mm respectively).

Runs 7, 8, and 9: Plain s&-For these three tests 35 feet (10.7 m) of the approach 3:l slope, the flat part of the crest, and all of the 4:l slope of the downstream bank were formed with the soil. For test 8. overtopping flow was about 87 (fts/s)/ft [8.1 (ms/s)/m]. The erosion volume was unexpectedly low compared to that for 40 (W/s)/ft [3.7 (ms/s)/m]. A soil test indicated a 102 percent maximum Proctor density rather than the 95-percent target density. Because of the overcompac- tion for the test 8, the embankment was reformed at 95 percent of Proctor maximum and over- topped at 87 (ft3/s)/ft for run 9.

Results

Two bed scour profiles (examples) are given on figure 14. The top profile is run 7 which was with plain soil embankment and without protective treatment effort other than compaction. The erosion volumes-determined from run 7-were used as base quantities to compare and normalize erosion volumes of the

other runs. The available erodible length for run 7 was considered to begin at the upstreamend of the 3:l approach soil slope and end at the downstream toe of the 4: 1 embankment slope. The bottom profile is run 5 which had the cobble roughened crest cap slope with downstream gabion protection. The available erodible length for run 5 was considered to begin at the downstream end of the gabion cover.

Table 5 contains sketches of the embankment treatments tested and a summary of model results.

l Column 1 shows the downstream slope sketches and test run numbers (circled).

l Column 2 lists the unit discharge being simulated. l Column 3 notes the total elapsed time of operation

before total erosion volume measurement. l Column 4 lists the relative volume erosion rate per

unit area along the erodible embankment length using run 7 as a comparison base.

l Column 5 lists the depth of embankment soil cover or erosion below an imaginary 2.5: 1 slope line and distance down the finished embankment slope measured from the downstream edge of the crest top. Negative values indicate scour below the 2.5:1 line. This base slope was selected because it is typical of Bureau embankment dams that are not rock protected.

l Column 6 lists the maximum scour depth relative to the finished embankment slope and its location on the slope.

Following are results and implications noted during the studies, tempered with experience from other sediment model studies and conversation with investigators experienced with overtopping occur- rences, and review of literature.

During test run 1, with a 6: 1 slope, flow rapidly transformed into a chute and pool mode. This type of flow is initiated by shallow bank surface waviness or by jets and eddies caused by flow around and over iso- lated projecting rocks in the fill material. Brush and trees would also cause local areas of increased scour. Scour holes tended to lii flow resulting in intermediate areas of less scour between areas of increased plunging flow scour. Road pavements, curbs, parapets, and bulging or sagging of top faces of gabion comparunents can initiate and affect location and amount of scour.

During test run 1, the smooth vertical walls of the model appeared to exaggerate scour, with deep holes in the embankment near the flume wails and the vertical curtain wall of the hard protective crest cap. Reservoir grain roughness and larger geometric bed form perturbations cause eddies that follow the flow over the upstream slope of the dam and stretch out parallel to the bed and intensify into vonices.

. .

Flow

,. . *

/rest Initial embankment profile

I - Hour scour profile

5-Hour scour profile

\ Reference slope

(a) Endmnkmant scour for run 7 used as base for run comparisons.

Flow -Crest /

Fixed Cobble roughened crest slope

/Gobions -Filter

lnitiol embankment profile

\ Reference slope

(b) Embankment scour for run 5.

F&we 14. - Typical scour of test embankment.

These vortices can cause local areas of relatively more intense bed shear. Some of the apparent wall effect may have been caused by upflow intensification. Crest end treatments not tested during these studies that contract the flow sideways would also cause vortices that would intensify local scour.

Generally, when Row passes a fixed bed to a soil bed or vice versa, scour occurs. This is caused by small boundary layer disturbances that form as flow ad- justs to abrupt changes in boundary roughness. An example of this type of scour is going from the hard crest cap to soil (test run 1). For test run 2, pea gravel was epoxied to the downstream sloping part of the hard crest cap simulating 3 to 6 inch (76 to 152 mm) cobble roughness increasing flow depth and damp- ening vortex action. This increase in roughness may have contributed to the significantly less, about one- half the erosion volume along available embankment length and the more uniform scour compared to test run 1. This effect also occurs for flow making the transition from gabions to soil.

Despite the previously discussed modeling limita- tions, velocity, depth, and discharge were scaled by Froude Law for comparison purposes. The scaled velocity for all test runs rapidly reached more than 30 ft/s (9.144 m/s). The general riprap design methods compared in reference (341 do not account for the combination of high velocity, steep downslope flow, nor shallow flow because they use uniform flow equations and entrainment functions developed empirically from uniform Row data. During test run 3, model riprap representing 6 to 24 inch-diameter (152 to 6 10 mm) stones became fluidized and eroded out immediately.

Gabions and mattress pods were modeled for test runs 4, 5, and 6. They were represented by cages or compartments formed with 4-inch (102-mm) mesh materials filled with angular rock up to 12inch (305-mm) maximum diameter and placed on a filter bed. During these tests, there was no indication that the gabions or mattress pods would be dislodged by the overtopping flow. It could not be determined if the embedded anchors were needed for the stabilii demonstrated. However, the author is unaware of manufacturer data for velocities greater than about 24 ft/s (7.3 m/s). As stated previously, the modeled velocities greater than 30 ft/s (9.144 m/s) occurred on the slope for all tests. On the 4: 1 slope the main erosion was near the downstream end of the protection. Whereas, for the 6:l slope the main scour hole was 36 feet (10.973 m) farther downstream. Comparing test runs 4 and 5 indicates that the effect of increasing the downslope from 6:l to 4:l increased the scour volume about five times. However, the maximum depth of scour was about 1-s times deeper. A minor part of the diierent scour locations

can be attributed to sagging and bulging of gabion compartment tops.

Erosion data for the 5hour test runs were used to observe if the rough Row later in the test had a greater or less transport rate than the earlier smooth flow. For these tests, the sediment transpon rate for the last 4 hours averaged about one-fourth of the rate for the first hour. Thus, it appears that the chute and pool mode flow reduced erosion relative to earlier smooth Row erosion. The percent of total volume eroded, computed from survey levels, did not significantly increase during the last 4 hours of test run 8. Possibly, not enough elevation measurements or cross sections were obtained with a surveyor’s level.

Model test results, of runs 8 and 9, having similar protective treatment, indicate that increasing standard Proctor compaction from 95 to 102 percent density reduced erosion by one-hatf.

Comparing results of test runs 7 and 9 showed that increasing unit discharge from 40 to 87 (fts/s)/ft r3.716 to 8.083 (m3/s)/m] resulted in about aper- cent increase in erosion.

Despite exercising care during field construction and inspection, low areas always exist along a dam crest. Low areas can occur after construction because of crest traffic, nonuniform settling, and lack of main- tenance. Erosion mode at a low point on the crest could depend on the shape of hydrograph during ovenopping. For example, with slow rising or low constant flow, the erosion may start and remain at one low point causing gulley type erosion, continuing on to breaching. tf flow rose fast enough, the effect of a low area could be drowned out and result in sheet flow erosion.

The reservoir overflow head on the crest should not be greater than about one-third the crest length in the direction of flow. This wwld cause a flow cavity at the upstream end of the crest. When head is less than about one-twetfth the crest length, friction af- fects flow considerably, undulation occurs and depth can approach critical more than once. Long-term flows outside the limit equation (12) increases risk of scour on the top of the crest.

OTHER POSSIBLE TREATMENTS NOT TESTED

Other possible treatments for downstream slope protection were discussed during meetings or found in the literature search. Some of these are expen- sive-material wise and labor intensive-but still

26

could be possible alternatives for treating existing srn8ll embankment dams. Possible treatments are:

1. Plastic sheets with thin expendable soil cover to protect them from ultraviolet and other physical damage. Another suggestion was plastic sheets rolled up under protective hoods and arranged so flow would roll them down the slope when needed. lt is possible that general bank waviness could rapidly c8use rippling or undulation of the plastic sheets resulting in tearing destruction.

2. Existing spillway capacity c8n be increased by hooding them so 8s to take advantage of additional head caused by siphon effect. The siphon must be designed properly to regulate air. Ermine [35] shows 8n example head-discharge curve for air regulated siphons. All or part of 8 spillway could be made into a siphon depending upon additional discharge needed. This method may be too costly.

3. Flow through and over mesh contained rock weirs and dams have been used and reported by AUStr8li8nS and New Zealanders. A summary of their experiences 8nd references 8re given by Thomas [36].

4. Reeves [37] discusses the use and economy of using roller-compacted COncrete to permit overtopping flow.

5. The U.S.S.R. developed protective blocks that interlock 8nd cause negative pressure underneath them so that they 8re increasingly sucked onto the embankment filter 8s flOW V8lOCity increases. Smith [38] model t8St8d and concluded “. . . that the compressive soil strength was the only limit to protection provided by the blOclcs.”

6. Another suggestion WBS the possibility of lii- ing the flow just below 8nd near the crest to flip it t0 8 downstream scour are8 Of leSS threat t0 the embankment.

BIBIJOGRAPHY

[l] Powledge, GA., 8nd R.A. Dodge, “Over- topping of Small Dams-An Alternative for Dam @fety,” Hydraulics and Hydrology in the Small Computer Age, Am8IiC8n Society of Ciiil Engi- neers Specialty Conference, August 1985.

[2] Miller, S.P., M. Hon-Yim Ko, and J. Dunn, “Em- bankment Overtopping,” Hydraulics and Hydrol- ogy in the Small Computer Age, American Society of Civil Engineers Specialty Conference, August 1985.

[3] Smerdon, E. T., 8nd R. P. BeaSley, “The Tractive Force Theory Applied to Stability of Open Chan-

nels in Cohesive Soils,” Research Bulletin 715, University of Missouri, 1959.

[4] Rouse, H., Elementary Mechanics of Fluids, Wiley, 1946,

[5] Kamphuis, J.W., “Determination of Band Roughness for Fixed Beds,” Journal of Hydraulic Research vol. 12, No. 2, 1974.

[6] Brown, B.J., and Y.H. Chu, “Boundary Effects of Uniform Size Roughness Elements in Two- dimensional Flow in Open Channels,” Miscella- neous Paper H-68-5, Corps of Engineers, Waterways Experiment Station, Vicksburg, Mis- sissipppi, 1968.

[7] Bathurst, J.C., R.M. Li, and D.B. Simons, “Re- sistance Equation For Large-scale Roughness,” Journal of Hydraulic Engineering, HY 12, Ameri- can Society of Civil Engineers, December 1981.

[8] Knight, D.W., and H.S. Patel, “Boundary Shear in Smooth Rectangular Ducts,” Journal Of Hy- draulic Engineering, vol. Ill, No. 1, January 1985.

[S]bos, M.G., “DiSCharge Measurement Btruc- tures,” Publication 20, ILRI, Defft Hydraulics Lab- oratory, The Netherlands, 1976.

[lo] King, H.W., “Handb00k of Hydraulics,” 3d 8d.. McGraw-Hill Book Co., Inc., 1939.

[ 1 l] Schlicting, “Boundary Layer Theory,” Perga- mon Press, 1955.

1121 Jacobs, W., “&tIfOrmung Eines Turulenten Geschwindigkeits-Profiles,” ZAMM, 19, 87- 100,1939.

[13] Gibbs, H.J., “A Study of Erosion 8nd Tractive Force ChaI’8Ct8riStiCS in Relation to Soil Mechan- ics Propenies,” 6um8u of Recl8matiOn Report No. EM-643, 1962.

[ld] Garlson, E.J., and P.F. Enger, “Studies of Trac- tive Forces of Cohesive soils in Earth Canals,” Bureau of Reclamation Report No. Hyd-504. 1962.

[ 151 Lane, E.W., “Progress Report on the Design of Stable Channels,” Bureau of Reclamation Report No. Hyd-352, 1952.

[16] Kelly, W.E., 8nd R.C. Gularte, “Erosion Resist- 8nce of Cohesive Soils,” American Society of Civil Engineers, HY 10, 1981.

[ 171 Kamphius, “Cohesive Material Erosion by Uni- directional Current,” JOUm8l of Hydraulic Engi- neering, vol. 109, No. 1, J8nU8v 1983.

27

[ 181 Simons, D.B., and F. Senturk, “Sediment Trans- port Technology,” WRP, Fort Collins, Colorado, 1977.

[19] Shields, A., “Application of Similarity Principles and Turbulence Research to Bedload Move- ment,” Translation CIT. Pasadena, California, 1936.

(201 Krishnamurthy, M., “Incipient Motion of Co- hesive Soils,” American Society of Civil Engi- neers Specialty Conference, Frontiers of Hydraulic Engineering, August 198 1.

(211 Carlson, E.J., “Critical Tractive Forces on Chan- nel Side Slopes in Coarse Noncohesive Material,” Bureau of Reclamation Report No. .Hyd-366, 1966.

(221 Brooks, N.H., Discussion, Journal of The Hy- draulics Division, American Society of Civil En- gineers, vol. 87, No. HY3, Proc. Paper 3529, May 1963.

(231 Simons, D.B., “Theory and Design of Stable Channels in Alluvial Material,” Ph.D. Dissertation, Colorado State University, Fort Collins, Colorado, 1957.

[24] Gessler, Johannes, “The Beginning of Bedload Movement of Mixture Investigated as Natural Ar- moring in Channels,” Translation T-5, Keck Lab- oratory, California Institute of Technology, Octobftr 1968.

[25] Graf, W.H., and G.C. Pazis, “Les Phenomanes de Deposition et D’erosion Dams un Canal Al- luvionnaire,” Journal of Hydraulic Research, vol. 15/2, 1977.

[26] Vanoni, V.A., “Sedimentation Engineering,” American Society of Civil Engineers, Manuals and Reports on Engineering Practices, No. 54,1975.

[27] Metha, A.J., T.M. Parchure, J.G. Dixit and R. Ariathurai, “Resuspension Potential of Depos- ited Cohesive Sediment Beds,” Estuarine Com- parisons, Academic Press, 1982.

[28] Kandiah, A., “Fundamental Aspects of Surface Erosion of Cohesive Soils,” Ph.D. Thesis, Uni- versity of California, Davis, 1974.

(291 Chen. Y.H., “Embankment Overtopping Test to Evaluate Damage,” American Society of Civil En- gineers Specialty Conference, Hydraulic and Hy- drology in the Small Computer Age, Lake Buena Vista, Florida, August 1985.

[30] Ariathurai, R., and K. Arulanadan, “Erosion Rates of Cohesive Soils,” Journal of Hydraulics Division, American Society of Civil Engineers, vol. 104, No. HY2, February 1978.

(311 Klein, S.J., “Similitude and Approximation The- ory,” McGraw-Hill Book Company, Inc., 1965.

[32] Bagnold, R.A., “Sediment Discharge and Stream Power-A Preliminary Announcement,” USGS Circular 421, 1960.

[33] Abt, S.R., and J.F. Ruff, “Estimating Culvert Scour in Cohesive Material,” Journal of Hydraul- ics Division, American Society of Civil Engineers, vol. 108, HYl, January 1982.

[34] Dodge, R.A., and C.J. Orvis, “Model Study and Riprap Design for Columbia River,” ASCE Spe- cialty Conference, Water Forum ‘8 1, Waterway Port, Coastal and Ocean Division, vol. II, August 1981.

[35] Ervina, D.A., “The Design and Modeling of Air- regulated Siphon Spillways,” Proceedings, I.C.E.., Part 82, B.H.R.A., May 1975.

[36] Thomas, H.H., “Flow Through and Over Rock- fill, Engineering for Large Dams,” part II, Wiley Interscience Pub., 1976.

[37] Reeves, G.N., “Planned Overtopping of Em- bankments Using Roller Compacted Concrete,” American Society of Civil Engineers Specialty Conference, Hydraulics and Hydrology in the Small Computer Age, Lake Buena Vista, Florida, August 1985.

[38] Smith, K.V.H., “Hydraulic Modeling of Materials for Protecting Earth Weirs,” International Con- ference, Hydraulic Modeling of Civil Engineering Structures, Coventry, England, September 1982.

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Documents

Denver Office August 1988 - Bureau of Reclamation · REC-ERG88-3 ‘ l I OVERTOPPING FLOW ON LOW EMBANKMENT DAMS - SUMMARY REPORT OF MODEL TESTS by R. A. Dodge August 1988 Hydraulics